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È Ä »¤¼ Ë ¾S¿8 ¸ óžô § ¸ * @ À . À . À »XÀ/¦É¾ ¸ ð » ð ô ó õ § » ð õ ó ô < Ì ¸ Ë ¾T Ë óžô µ ¾ ð § ð ì » ð ô ó õ § » ð õ » À;¦É¾ ¸ ð ó ô < ¾ ð Ñ , * ¨ ì Þ 74 Now we are ready to approximate the integral in à by the stationary phase method. The general formula states that for ÔC¤ÔED F , 'H ¸ à’¾¥)JI . ¨ Ã G ¸ à K ¾ãú$&(‡À (A-16) H ¸ à K ¾¥) ¼ Å à>G ¸ à ¾ãú$'&(‡À ¸ àK%¾ is the first-order stationary point of H , i.e., H >H where K ¸ our à ¾S¿øÑ ¸ »¤¼ Ë ¼bà ¾ case, , K ¿O¤ , H à ¸ ¤Š¾S¿øÍ à H à ¾!¿L¤ ¸ ¸ and à K ¾NM H K ¿ À µP Ñ and H ¸ ¤Š¾œ¿]Ñ . Ñ ÿ Applying all the results above to the integral in à in equation (A-7), we find ¿L¤ . In £¢ ú%$'&(‡À ·5¸ »¤¼¢ ¾œ¿Q ¸ ¢ ¾ Substituting the kernel È ¨ Ã Ä Å Ë Ì ¸ Ë ¾ È zÍ£BÑ À ê Ç Í Û ÑR , Ñ given by equation (A-13), we finally arrive at ÑS) Þ ¢ (A-17) ¢ ·5¸ »¤¼¢ ¾ ¿ ¸ ¢ ¾ U: § ¸ ¨ Ã ÄÆÅ µ ê Ç ¾ óžõ Ñ , * ¨ (A-18) ÀV* ú%$'&(‡À ¢ ÑS) ¼ ç ð ê Ç = or, equivalently in time-domain, (A-19) »¤¼%½Š¾œ¿ µ ·5¸ ¨ Ã Ä Å , Ë Ì ¸ Ë ¾ È ðW X ¸ »¤¼ Ë ¾!YZ¢ Ï ¸ ½ÐÀÒÑ ¸ »¤¼ Ë ¼ Ç ¾¾'¼ where denotes the depth-reverse half-derivative, which is equivalent to multiply by ¢ in frequency domain, and where the 2.5D weight function for constant velocity YR¢ is given by , À (A-20) À;* : ðW X ó÷õ § ¸ ê Ç ¾ ¸ * @ À ê Ç = ç ð
Wave Inversion Technology, <strong>Report</strong> No. 4, pages 75-86 Offset-dependent resolution of seismic migration J. Schleicher and L.T. Santos 1 keywords: resolution, seismic migration, Fresnel zone ABSTRACT In this work, we study the resolving power of seismic migration as a function of sourcereceiver offset. We quantify horizontal resolution by means of the region around the migrated reflection point that is influenced by the migrated elementary wave. To obtain an estimate for the mentioned zone of horizontal influence after migration, we investigate the migration output at a chosen depth point in the vicinity of the specular reflection point, i.e., when the output point is moved along the reflector. We find that the region of influence is well approximated by the difference between the time-domain Fresnel zone and its paraxial approximation. The width of the spatial resolution resulting from migration of the reflection event is compared with the resolution predicted from theoretical ray-theory formulas for various data sets with different offsets. It is to be remarked that the above resolution is reached only with perfect, noise-free, correctly sampled, unbiased data. INTRODUCTION Seismic resolution after depth migration has been theoretically discussed by various authors (Berkhout, 1984; Beylkin, 1985; Cohen et al., 1986; Bleistein, 1987). A recent comprehensive study on the subject was carried out in Vermeer (1999), where additional references on the subject can be found. It is widely accepted among geophysicists that 'depth migration reduces the Fresnel zone'. Although this is a very sloppy expression, because the Fresnel zone is a fixedsize frequency-dependent quantity associated with the reflected ray, we will see in this section that there is a lot of truth in it. We discuss horizontal resolution in a completely analogous manner to the discussion of the pulse stretch in Tygel et al. (1994) that is closely related to vertical resolution. Note that we implicitly define now resolution in a slightly different way from what is usually done in the literature. Conventionally, resolution is quantified 1 email: js@ime.unicamp.br 75
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Wave Inversion Technology WIT Annua
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Wave Inversion Technology WIT Wave
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Copyright c 2000 by Karlsruhe Unive
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i TABLE OF CONTENTS Reviews: WIT Re
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Wave Inversion Technology, Report N
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3 theory. It relates properties of
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Imaging 5
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8 two hypothetical eigenwave experi
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7 ¢¥£§¦©¨ ; < calculate sear
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7 7 searches ¡ HNJOL for ¢IHNJML
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14 Trace no. 1000 1500 2000 0.5 1.0
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16 CDP number 3100 3000 2900 2800 2
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18 REFERENCES Höcht, G., de Bazela
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20 INVERSION BY MEANS OF CRS ATTRIB
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22 For synthetic data, it is easy t
Page 36 and 37: — J J J J J J G J - J J G
Page 39 and 40: Wave Inversion Technology, Report N
Page 41 and 42: · · v 0 B 0 £ & Ä Ã & v
Page 43 and 44: 31 Dürbaum, H., 1954, Zur Bestimmu
Page 45 and 46: Ã Ø Ì 0 0 Ì 4 0 Ã 0 G ' 4
Page 49 and 50: 0 0 ˜ ” Z ˜ ” Z 4 4
Page 51 and 52: 39 strategy by combining global and
Page 53 and 54: 41 elled data is presented in Figur
Page 55 and 56: 43 0.2 Distance [m] 1000 1500 2000
Page 57 and 58: 45 In Figure 10 we have the optimiz
Page 59: 47 Gelchinsky, B., 1989, Homeomorph
Page 62 and 63: § ¢ ø ø for a paraxial ray, ref
Page 64 and 65: Œ § … Z ‹ Z § õ ø \ ø §
Page 66 and 67: 54 As stated in Hubral and Krey, 19
Page 68 and 69: § ó § 56 sequence. We made tes
Page 70 and 71: 58 precision of the modeled input d
Page 72 and 73: 60 Cohen, J., Hagin, F., and Bleist
Page 77 and 78: £ 65 The details of the theory inv
Page 79 and 80: 67 of the figure. On both sides, a
Page 81 and 82: 69 (a) (b) Depth (m) 600 800 Depth
Page 83 and 84: 71 Langenberg, K., 1986, Applied in
Page 85: È Û Ñ¿ = ¨ Í Ñ>* = ¿ + ð
Page 89 and 90: g ý g [ ^ â g [ g g g g g â h [
Page 91 and 92: g § » ¹ [ƒŽ g 79 We now subst
Page 93 and 94: ê 81 ing was realized by an implem
Page 95 and 96: ˆ 83 -0.05 -0.1 Amplitude -0.15 -0
Page 97: 85 on a second-order approximation.
Page 100 and 101: 88 kinematic (related to traveltime
Page 102 and 103: ¨ º Ý È · § À · Á · Á ¼
Page 104 and 105: ° ´ ´ ã ° ¼ ´ ´ þ Ò Ò ¥
Page 106 and 107: 94 1 taper value ksi1 0 ksi2 Figure
Page 108 and 109: 96 0 1 2 3 4 5 Depth [km] CMP [km]
Page 110 and 111: 98 CONCLUSION As a generalization o
Page 112 and 113: 100 weight during the stacking proc
Page 114 and 115: 102 are not illuminated for every o
Page 116 and 117: 104 obtained from Zoeppritz' equati
Page 118 and 119: 106 PreSDM of Porous layer 0.358 re
Page 120 and 121: 108 REFERENCES Gassmann, F., 1951,
Page 122 and 123: 110 et al., 1992), Hanitzsch (1997)
Page 124 and 125: 112 consecutive wavefronts). Howeve
Page 126 and 127: 114 Numerical example We test the W
Page 128 and 129: 116 Versteeg, R., and Grau, G., 199
Page 130 and 131: 118 indicator. To extract elastic p
Page 132 and 133: € b b 120 S where is the position
Page 134 and 135: É É É É 122 The À constant (da
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124 0 Distance [ m ] 1000 2000 3000
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126 Kosloff, D., Sherwood, J., Kore
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128
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130 penetration distances and a pot
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and » ˆ Ö Ö is the scalar hydra
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Å éÙó ó ».-.‹œì/-jì¨‹
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136 TRIGGERING FRONTS IN HETEROGENE
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Ö ˆ Ö “P£'ˆ é ˆ ó,Lˆ¨
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Ö Ö Ê » Ø Ö » Ö Ê Ê Ê Ö
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142 300 250 200 a) eikonal equation
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144 Shapiro, S. A., and Müller, T.
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» i » m ×]Ú ‘Œƒšj'Ÿlk hM
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148 RESULTS Soultz-sous-Forêts Inv
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…„ƒ 150 for the smaller ellips
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152 Figure 5: The cloud of events f
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156 straightforward. The new approa
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158
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160 1982). There is also the so-cal
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Ï u u ¬ 164 To summarize, we deri
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Ý ì á ä é å¤æDçzè ä é
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ø M ã = ã Ù ø ä Ú ã Ù ø
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ã Q c ã ä 170 a=10m; std.dev.=8%
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172 Frankel, A., and Clayton, R. W.
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174 It is well-known that inhomogen
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ã ä ä ŸSŸ S ¡S¡ ©©¨¨ æ
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Kneib, 1995 285-6000 superposition
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180 parameter: Hurst coefficient pa
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182 REFERENCES Bourbié, T., Coussy
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184 1999) in multiple fractured med
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j jÆÅÇ[È”u j jÎÅÇ[È”uw
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190 Davis, P. M., and Knopoff, L.,
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192 properties from the measured se
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194 discussion of these problems ca
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196 Altogether, close to 260 shotpo
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198 Time (ms) 0 2 4 6 8 10 12 14 16
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200 of groundwater. ACKNOWLEDGEMENT
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202
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204 the wavefront curvature matrix
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û ò é from ã äñ to ã î§ñ
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é 208 Table 1: Median relative err
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210 Table 2: Median relative errors
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212 CONCLUSIONS We have presented a
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214 PUBLICATIONS Previous results c
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æVU ìÿå T ü ýWYXZX[ 9\]9^\ Þ
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218 weight functions from traveltim
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220 REFERENCES Bleistein, N., 1986,
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226
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228 In this paper, we present a mor
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L • MM+,ON N N Ž ú R N N N ú
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232 Figure 3: The evolution of a WF
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234 Let us analyze next the computa
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236 CONCLUSIONS We have presented a
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238
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’ Ç r 240 traveltimes, a computa
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Ž Ž Ã Ž 242 Other two approxima
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“ Ã Æ ³ 244 0 0.5 [km] 0 0.5 1
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î r † Û(Û Ü Œ U Ú Ý Ü Û
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248 Ettrich, N., 1998, FD eikonal s
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ø Ð ¡ Ð÷ö Ð'ø ú Ð ¡ Ð'
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‹ 256 transport equation. Neverth
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258 tivalued geometrical spreading
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260 .
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262
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… " ¯ ü ý +- 4=° ü ü +
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ö ô æ º Ò x ¹ v ý ñ
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268 NUMERICAL RESULTS We constructe
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¹ õ ' ' -š' ù ¹ ù ' ' -š
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£ -š' - 272 Ô- —- Figure 6: C
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274 REFERENCES Aldridge, D. F. (199
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276
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278 3. 4. True-amplitude imaging, m
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280 Research Group Hamburg (Gajewsk
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282 Stefan Lüth Robert Patzig Refr
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284 Landmark Graphics Corp. 7409 S.
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286 TotalFinaElf Exploration UK plc
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288 as research assistant at Geoeco
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290 Ingo Koglin is a diploma studen
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292 Matthias Riede received his M.S
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294 Svetlana Soukina received her d
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296 Yonghai Zhang received the Mast
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Annual Report 2000 - WIT

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